# Structure of Waiting line Models

There are four basic waiting line structures that describe the general conditions at a service facility. The simplest structure is our basic module. It is called the single server case. There are many examples of this simple module: the cashier at a restaurant, any single window operation in a post office or bank, or a one-chair barber shop.

If the number of processing stations is increased but still draws on a single waiting line, we have the multiple servers case shown. A post office with several open windows but drawing on a single waiting line is a common example of a multiple server waiting line structure.

A simple assembly line or a cafeteria line has, in effect, a number of service facilities in series and is an example of the single server.

Finally, the multiple severs in series case can be illustrated by two or more parallel assembly lines. Combinations of any or all of the basic four structures can also exist in networks in very complex systems.

The analytical methods for waiting lines are divided into two main categories for any of the basic structures shown, depending on the size of the source population of the inputs. When the source population is very large and, in theory at least, the length of the waiting line could grow without fixed limits, the applicable models are termed infinite waiting line models. On the other hand, when arrivals come for a small, fixed size population, the applicable models are termed finite waiting line models. For example, if we are dealing with the maintenance of a bank of 20 machines and a machine breakdown represents an arrival, the maximum waiting line is 20 machines waiting for services, and a finite waiting line model is needed. On the other hand, if we operated an auto repair shop, the source population of breakdowns is very large and an infinite waiting line model would provide a good approximation. We will discuss both infinite and finite models.

Here are other variations in waiting line structures that are important in certain applications. The queue discipline describes the order in which the units in the waiting line are selected for service. We imply that the queue discipline is first come first served. Obviously, there are many other possibilities involving priority systems. For example, in a medical clinic, emergencies and patients with appointments are taken ahead of walk in patients. In production scheduling systems there has been a great deal of experimentation with alternate priority systems. Because of the mathematical complexity involved, Monte Carlo simulation has been the common mode of analysis for systems involving queue disciplines other than first come first served.

Finally, the nature of the distributions of arrivals and services is an important structural characteristic of waiting line models. Some mathematical analysis is available for distribution that follows the Poisson or the Erlang process (with some variations) or that have constant arrival rates or constant service times. If distributions are different from these mentioned or are taken from actual records, simulation is likely to be the necessary mode of analysis.

Infinite waiting line models:

We will not cover all the possible infinite waiting line models but will restrict our discussion to situations involving the first come first served queue discipline and the Poisson distribution of arrivals. We will deal initially with the single server case (our basic service facility module); later we will also discuss the multiple server case. Our objective will be to develop predictions for some important measures of performance for waiting lines, such as the mean length of the waiting line and the mean waiting time for an arriving unit.